is the category whose objects are lattices and whose morphisms are lattice homomorphisms, that is functions which preserve finitary meets and joins (equivalently, binary meets and joins and the top and bottom elements). is a subcategory of Pos.
is given by a finitary variety of algebras, or equivalently by a Lawvere theory, so has all the usual properties of such categories. By general abstract nonsense, the free lattice on a set exists, but it is not easy to describe in general.
The results about the free lattice on generators may be found in Section 4.6 of Stone Spaces, which sites Skornjakov's Elements of Lattice Theory.